Microwave Ring Circuits and Related Structures phần 2 doc

30 ANALYSIS AND MODELING OF RING RESONATORS

G1

G2

l

1

l

2

l=l1

+l2

-z1

-z2

z1,2=0

V1

V2

I

1

I2

I

V

(a)

G1

G2

l

1

l

2

l=l1+l2

-z1

-z2

z1,2=0

V1

V2

I

1

I2

I

V r

(b)

FIGURE 2.15 The configurations of one-port (a) square and (b) annular ring

resonators [10].

TABLE 2.4 A Comparison of Table 2.3 and the

Theoretical Results from (upper) the Transmission Line

Method and (lower) the Magnetic-Wall Model

Frequency Error (%)

Circuit n = 1 n = 2

1 0.79 0.60

2 0.28 0.49

3 0.56 0.28

Frequency Error (%)

Circuit n = 1 n = 2

1 0.78 0.89

2 0.07 0.37

2 0.63 0.38

is considered to be a transmission line. z1 and z2 are the coordinates corre￾sponding to sections l1 and l2, respectively.The ring is fed by the source voltage

V at somewhere with z1,2 < 0. The positions of the zero point of z1,2 and the

voltage V are arbitrarily chosen on the ring.

For a lossless transmission line, the voltages and currents for the two

sections are given as follows:

(2.65a)

(2.65b)

where V+

oe-jbz

1,2 is the incident wave propagating in the +z1,2 direction,

V+

oG1,2(0)ejbz

1,2 is the reflected wave propagating in the -z1,2 direction, G1,2(0) is

the reflection coefficient at z1,2 = 0, and Z0 is the characteristic impedance of

the ring.

When a resonance occurs, standing waves set up on the ring. The shortest

length of the ring resonator that supports these standing waves can be

obtained from the positions of the maximum values of these standing waves.

These positions can be calculated from the derivatives of the voltages and

currents in Equation (2.65). The derivatives of the voltages are

(2.66)

Letting , the reflection coefficients can be found as

G1,2(0) = 1 (2.67)

Substituting G1,2(0) = 1 into Equation (2.65), the voltages and currents can be

obtained as

(2 .68a)

(2.68b)

Based on Equation (2.68), the absolute values of voltage and current stand￾ing waves on each section l1 and l2 are shown in Figure 2.16.

Inspecting Figure 2.16, the standing waves repeat for multiples of lg/2 on

the each section of the ring. Thus, to support standing waves, the shortest

length of each section on the ring has to be lg/2, which can be treated as the

fundamental mode of the ring. For higher order modes,

l n for n = 1, 2, 3, . . . (2.69) g

1 2 2 , = l

I z j V

Z z o

o

12 12 12

2 ,, , ( ) = - sin( ) +

b

Vz V z 12 12 12 ,, , ( ) = 2 o cos( ) + b

∂ ( )

∂ =

=

V z

z z

12 12

1 2 0 1 2

0 , ,

, ,

∂ ( )

∂ =- - ( ) ( ) V z + -

z

jV e e o

12 12 jz jz

1 2

1 2 12 12 0 , ,

,

, , , b b b G

I z V

Z

e e o

o

jz jz

12 12 12 12 12

,, , 0 , , ( ) = - ( ) ( ) + - b b G

V z Ve e o

jz jz

12 12 12 12 12

,, , 0 , , ( ) = + ( ) ( ) + - b b G

TRANSMISSION-LINE MODEL 31

where n is the mode number. Therefore, the total length of the square ring

resonator is

l = l1 + l2 = nlg (2.70)

or in terms of the annular ring resonator with a mean radius r as shown in

Figure 2.15b,

l = nlg = 2pr (2.71)

Equation (2.70) shows a general expression for frequency modes and may be

applied to any configuration of microstrip ring resonators, including those

shown in [28, 29].

2.4.7 An Error in Literature for One-Port Ring Circuit

In [11], one- and two-port ring resonators show different frequency modes. For

a one-port ring resonator, as shown in Figure 2.17a, the frequency modes are

given as

n = 1, 2, 3, . . . (2.72a)

f (2.72b) nc

r o

eff

= 4p e

2

2

p

l

r n g =

32 ANALYSIS AND MODELING OF RING RESONATORS

l

1

l

2

l=l1

+l2

-z1

-z2

z1,2=0

V1

V2

I1

I2

I

V 2 2 V z( ) 2 2 I z( )

-z2

1 1 V z( ) 1 1 I z( )

2

-lg -lg -z1 -z1=0

-lg

2

-lg -z2=0

FIGURE 2.16 Standing waves on each section of the square ring resonator [10].